Geometric Series

Sum of geometric series

Geometric series is very important in analysis. It denition is

$\sum _{n=0}^{\infty}(z)^n$

Calculate this infinite sum is easier as seems if we note following

$S_{n} = \sum _{n=0}^{\infty}(z)^n = 1 + z + z^2 + ... + z^n \Rightarrow zS_{n}= z + z^2 + ... + z^{n+1}$

this implies

$S_{n}(1-z) = 1 + z + z^2 + ... + z^n - (z + z^2 + ... + z^{n+1}) = 1 + z^{n+1}$

thus

$S_{n} = \frac{1 + z^{n+1}}{1-z}$

Now is easy taking limit when $n\mapsto \infty$

$\sum _{n=0}^{\infty}(z)^n = \frac{1}{1-z}$

We apply the ratio test

$\lim_{n\to \infty} |\frac{S_{n+1}}{Z_{n}}| = \lim_{n\to \infty} |\frac{x^{n+1}}{z{n}}| = |z|$

Thus, geometric series converges if |z| < 1 and diverges if |z| > 1